What is different between Bernoulli's equation and Continuity equation?

Answer 1

Answer First, we use the continuity equation to relate the two velocities. The two areas are related by A1 = 4A2 (note that it's 4, not 2: if the radius doubles, the area quadruples.) Thus the continuity equation says that v1A1 = v2A2 so that v2 = 4v1 (the uid in the larger pipe ows slower, and with greater pressure). Now using the above equation gives P1 􀀀 P2 = 1 2 (v2 2 􀀀 v2 1) 100 Pa = 1 2 (1000 kg=m3)(16v2 1 􀀀 v2 1) Solving for v1 gives v1 = s 100 (500)(15) m=s = :12m=s To get the full Bernoulli's equation, we just need to include gravity like we did before. Now let's consider a pipe which goes up and downhill. Then we need to do work to get the water to go uphill as well, from y1 to y2. The potential energy is as always mgh. We then have (P1 􀀀 P2)V = 1 2 mv2 2 􀀀 1 2 mv2 1 + mg(y2 􀀀 y1) 3 Rearraging again gives Bernoulli's equation: P1 + 1 2 v2 1 + gy1 = P2 + 1 2 v2 2 + gy2 If you set the velocities equal to zero like we were doing last time, then this equation reduces to our earlier one. There are many consequences of Bernoulli's equation. In fact, Bernoulli's equation also applies to air. It's why planes y, and curve balls curve. The basic idea is that when air is moving faster, its pressure is lower. Thus if you manage to have an object like a Baseball or a plane wing with dierent pressures on dierent sides, the object will feel a net force. A spinning baseball drags some of the air with it. This means on opposite sides, the air is moving at dierent speeds. This results in a change in pressure, and makes the ball move sideways. The shape of a plane wing causes the air rushing past to go faster on top. This lowers the pressure and hence causes the wing to lift up. 4